GBQ Question 42

[stormin]

At what time on a watch are the hour and minute hands placed so that in terms of minutes past 12 o'clock one is exactly the square of the distance of the other?
I do not want the answer, but I would like help in understanding the question. Any help in that respect would be much appreciated.

[Solon]

Stormin,

For each minute the hour hand moves through one minute. (making a full circle in sixty minutes.

Suppose the hour hand is as 12:05. The distance is 5 mins. The minute hand therefore needs to be at 5x5 minutes (i.e. 5 squared) past the hour.

[crofter]

I don't think the question really means "distance" otherwise the length of each hand comes into play! I think the reference is purely in terms of "TIME".

When it is 25 past 5 exactly, where are the hands pointing? Is one hand pointing to the squared value of the other?

I'd like to feel that fractional parts could appear in the answer! If so, then I would expect seven possible solutions!

[swannbaker]

You need to consider all the squares , technically , between 1 and 3600 ( 60 squared . For example , 100 , being a square would put the minute hand on the 8 on the clock i . e . 100 minutes past 12 . In which case , where would the hour hand be ?.

[swannbaker]

I should have added that if the hour hand is on the 2 , then you have your answer .

[Solon]

Crofter,

Assuming we have a 'conventional' watchface the length of the two hands are irrelevant because the hour hand moves 30 degrees each hour whilst the minute hand moves 360 degrees. The ratio of the distance from 12:00 will always be the same whether the hands are 2mm or 2 mile long.

[crofter]

Solon - I was trying to make the very same point! The question uses the phrase "the square of the DISTANCE of the other". In this case, the choice of the word "distance" to denote "angles" was not ideal

[swannbaker]

The word "distance" does seem OK . To satisfy the question for instance , if the minute hand has travelled a distance of 225 minutes , the hour hand must have travelled through a distance of 15 minutes i . e . the square root of 225 . The answer to the question is , of course , a lot easier to work out than we have made it appear .

[Solon]

Afternoon Swannbaker,

We should clarify that if the hour hand has travelled 15 minutes (e.g. 3 hours) the minute hand must have travelled only 180 mins.

For the minute hand to travel 225 minutes the hour hand will have moved 19 minutes n'est pas?

Must admit when I first read the question I fell for the trap of thinking of the 12:00 as the 'top' of 360 degrees so didn't scibble anyting om the back of the envelope past 13:00.